reserve a,b,c,d for Real;
reserve z,z1,z2 for Complex;

theorem Th65:
  |.z1*z2.| = |.z1.|*|.z2.|
proof
  set r1 = Re z1, r2 = Re z2, i1 = Im z1, i2 = Im z2;
A1: 0<=r1^2 + i1^2 & 0<=r2^2 + i2^2 by Lm1;
A2: (Im(z1*z2))^2 = (r1*i2 + r2*i1)^2 by Th9
    .= 2*(r1*r2)*(i1*i2) + ((r1*i2)^2 + (r2*i1)^2);
  (Re(z1*z2))^2 = (r1*r2 - i1*i2)^2 by Th9
    .= (r1*r2)^2 + (i1*i2)^2 + - 2*(r1*r2)*(i1*i2);
  then (Re(z1*z2))^2+(Im(z1*z2))^2 = (r1^2 + i1^2)*(r2^2 + i2^2) by A2;
  hence thesis by A1,SQUARE_1:29;
end;
